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Running head: ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 1
The Effect of Item Format, Question Type, Personal Interest, and Gender on Algebra
Achievement on a Computer-Based Assessment
Michael Mazzarella
George Mason University
Fairfax, Virginia
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 2
Abstract
Previous research shows that item format, question type, interest, and gender have effects
on math achievement, but few studies have focused on these variables together on a computer-
based assessment. The current study will analyze the responses of 225 high school Algebra II
students on a computer-based math assessment with different question types and item formats, as
well as a survey measuring their personal math interest (gender?). The test results will be
analyzed using Classical Test Theory (CTT) and Item Response Theory (IRT) statistics, while
other comparisons will be made using inferential statistics, such as correlation and ANOVA.
Keywords: item format, technology-enhanced, interest, attainment value, perceived task
difficulty, knowledge transfer, information processing.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 3
The Effect of Item Format, Question Type, Personal Interest, and Gender on Algebra
Achievement on a Computer-Based Assessment
Within the last decade, many schools and school districts across the country have begun
administering end-of-the-year tests and other standardized assessments on the computer. Taking
these exams on the computer can impact how students perform on tests, even when the content is
the same as that of a paper-and-pencil test (Scherer & Siddiq, 2015). This change can drastically
impact student achievement, especially for high-stakes assessments. For example, in the state of
Virginia, the Standards of Learning (SOL) Assessment has been well-known by teachers,
students, and parents alike. Passing this assessment in almost all of the core high school subjects
is required for graduation (Virginia Department of Education, 2012). In the 2011-2012 school
year, the state of Virginia changed the format of all of the SOL exams in order to make them
more technology-based. All math SOL assessments now include fill-in-the-blank, drag-and-drop,
and graph plotting questions, instead of only multiple choice questions. In the first year, passing
rates for all math SOL exams significantly dropped. Many who are familiar with the test,
however, wonder whether this drop can be attributed to a change in item format, the introduction
to computer-based testing, or an increase in difficulty of the exam (source?).
Information Processing Theory
The cognitive processes that students undergo while taking assessments can be partially
explained by the information processing theory (IPT). According to Mayer (2012), IPT can be
defined as humans creating mental representations and applying cognitive processes to them.
Martin (2004) distinguished between two interpretations of information processing: literal, which
is the processing of information, and constructivist, which is the construction of knowledge. In
an earlier work, Mayer (1995) introduces the metaphor of a computer to explain IPT: Human
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 4
minds are like processing systems, which have memory storage and control processes. Figure 1,
found in Appendix A, illustrates the basic model of information processing. The model shows
how sensory, working, and long-term memory work together to select, organize, and integrate
information. In the context of assessments, the goal of creating a good assessment is to properly
test what students know. A test will be more reliable when it is based on what is known about
human information processing (Mayer, 2011).
Several studies have related the IPT to assessments and mathematics. Raghubar, Barnes,
and Hecht (2010) reviewed the prior research relating working memory and mathematics. They
found that working memory is involved in arithmetic, but the amount of working memory used
during math processing may vary depending on age or strategy use. Furthermore, the way that
the task is presented to the individual is also related to working memory (Raghubar, Barnes, &
Hecht, 2010). This conclusion holds true for computer-based assessments, and it is also worthy
to note that working memory and math processing is also impacted by the task difficulty and
question type (Goldhammer, Naumann, Stelter, Toth, & Rolke, 2014). These findings are
relevant to the current study because the study will analyze computer-based math problems of
varying types and formats.
Assessments: Item Format
In the present study, item format refers to the ways in which students can respond to the
prompts. Some item formats include multiple choice, true or false, selecting multiple correct
answers, and fill-in-the-blank. Many research studies have been conducted regarding the
relationship between item format and achievement. Ozuru, Best, Bell, Witherspoon, and
McNamara (2010) measured achievement on a reading assessment. Items were separated into
multiple choice and open ended questions. Overall, students scored the multiple choice questions
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 5
correctly more often than the open ended questions. The average score for the multiple choice
section without and with available text was 68% and 79%, respectively, while the average score
for the open ended section without and with available text was 48% and 60%, respectively
(Ozuru, Best, Bell, Witherspoon, & McNamara, 2010). The large gap between the scores of the
multiple choice and open ended questions suggests that students think about each type of
question differently, which affects achievement on assessments.
Not only can different item formats produce different achievement scores, but they can
result in different strategies used. Katz, Bennett, and Berger (2000) had students record the
strategies used for each item on an assessment with multiple item formats. Traditional strategies
included using formulas and solving algebraic equations, while nontraditional strategies included
estimation and guess-and-check. Results showed that while item format for some questions
changed their levels of difficulty, the strategies used for different questions did not affect the
achievement (Katz, Bennett, & Berger, 2000).
Item format is particularly important for computer-based exams. Jodoin (2003) studied
responses on an engineering exam that used multiple choice and “innovative” items, such as
drag-and-drop and selecting multiple answers. Results showed that examinees answered multiple
choice items with more accuracy, even when items of different formats were asking about the
same subject (Jodoin, 2003).
It is important to note that measures with different item formats can threaten validity in
several ways. For example, in one of the first articles looking at computer-based assessments
with different formats, Martinez and Bennett (1992) analyzed several types of math skills (e.g.
algebraic reasoning, computer science) that were tested using different item formats.
Psychometric analysis determined that there was little discrepancy between computerized raters
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 6
and human raters, but in some cases, scores differed by up to 1.2 points on a 16-point scale
(Martinez & Bennett, 1992). Because of this small difference, computerized grading gained more
trust; nevertheless, even a small difference such as this highlights the importance of calculating
validity and reliability of computer-based assessments.
In a similar article, Pomplun and Omar (1997) outlined four different threats to the
validity of an assessment: lack of familiarity of the item format, omitting alternatives,
dependency among alternatives, and guessing. In their study, two specific types of item formats
used were “multiple mark” (a multiple choice question with more than one correct answer) and a
“multiple true-false question” (similar to multiple mark, but only with T/F as possible choices).
The results of this article stated that omitting answers or not following directions did not
seriously threaten validity, but students tended to leave choices blank instead of guessing
(Pomplun & Omar, 1997).
Assessments: Question Type
Assessing mathematics knowledge is a broad idea that can be categorized into multiple
aspects. Thus, it is important to study differences in the ways that questions are asked on math
assessments. In particular, two such types of math problems are computational (i.e.
straightforward) problems and word problems. Fuchs et al. (2008) studied whether or not
different aspects of cognition (e.g. language, concept formation, and working memory) were
used in different types of problems. Results showed that correlations between computational and
problem-solving skill was only moderate. Cognitively, processing speed was highly correlated
with computational skill, but not with problem-solving skill. Working memory, on the other
hand, was more highly correlated with problem-solving skill than with computational skill
(Fuchs et al., 2008). The same study also compared demographics to achievement on different
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question types. For example, poverty and race had little effect on difficulty with computational
problems, but those who had difficulty with solving word problems were poorer and more likely
to be African-American (Fuchs et al., 2008). Therefore, it is certainly worthwhile to study
question type as it relates to math achievement.
When studying question type, one must take into consideration the construct of
knowledge transfer. Knowledge transfer can be defined as the ability to relay knowledge that
students learned one way to a task presented in a different way (Belenky & Nokes-Malach,
2013). In the present study, knowledge transfer is relevant regarding question type because
students may have learned a math procedure in a straightforward manner, but may not be able to
apply that knowledge to a word problem, for example. Knowledge transfer also relates to
assessments from an information processing viewpoint. Working memory is necessary for
students to be able to transfer, organize, and apply their knowledge to a given task (Belenky &
Nokes-Malach, 2013). Day and Goldstone (2012) state that despite cognitive load or item
difficulty, student transfer is high when students are shown several examples of the problem.
They go on to claim the following: “Contextual similarity between the situations themselves
seems to play a much larger role in determining whether transfer will actually occur” (Day &
Goldstone, 2012, p. 155).
Kramarski, Weiss, and Sharon (2013) experimented with an intervention that attempted
to increase students’ transfer abilities among math tasks by increasing self-regulation. The
researchers gave students a survey measuring three aspects of self-regulation (planning,
monitoring, and evaluation), procedural knowledge algebra tasks, and verbal algebra problem
solving tasks, which they classified as “long-term transfer to novel tasks.” The results showed
that while there was no significant difference between the two learning approaches for the
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 8
procedural knowledge tasks, there were significant differences between the two learning
approaches for the problem-solving tasks, including all three specific types of mathematical
categories (algebraic, number sense, and visualization; Kramarski, Weiss, & Sharon, 2013). In
terms of self-regulation, those who were part of the intervention group tended to be higher in
self-regulation, which led to high scores on mathematical tasks that were classified as “far
transfer” tasks (number sense and visualization; Kramarski et al., 2013). Overall, knowledge
transfer is another construct to consider in all aspects of assessments, especially question type
and item format.
Computer-Based Assessment
As mentioned, many standardized tests are now computer-based, which prompts research
on the effects of technology on assessments, especially those in mathematics. In addition to the
studies mentioned previous regarding item format and question type, other studies focus on the
effects of computer-based assessments. Burns, Klingbeil, and Ysseldyke (2010) looked at the
effects of a technology-enhanced formative evaluation (TEFE) system on elementary school
math scores in four states. Results shows that schools that had been using the TEFE system for
more than five years reported higher math scores than schools that had been using it for between
one and four years; schools who had been using it for between one and four years scores higher
than schools who did not have a TEFE system (Burns et al., 2010).
Threlfall, Pool, Homer, and Swinnerton (2007) created a study in England that compared
student achievement on math assessments on paper-and-pencil to student achievement on
mathematics assessments on a computer using the same exact questions. Results showed that out
of seven mathematical categories, students who took the computerized version of the exam
scored higher on five of them. The authors attribute this difference in scores to the interactive
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 9
tools that students could use on the computer (i.e. moving items across the screen; Threlfall et
al., 2007). A similar study compared the results of a paper-and-pencil test to those of a computer-
based test in an American elementary school classroom. The researchers found that some
computer-based measures tended to be significantly less reliable than their paper-and-pencil
counterparts. Furthermore, there were inconsistencies between the multiple computer-based
measures (Shapiro & Gebhardt, 2012). This suggests that changing an assessment from paper-
and-pencil to computer-based requires analysis to ensure that the validity is still strong, even if
the same paper-and-pencil test was found to be valid and reliable.
While there is much research about computer-based instruction, less research exists about
the effects of computer-based delivery of an assessment on student achievement. The current
study hopes to extend this research and understand whether or not technology is an effective tool
in math assessment delivery.
Interest
While knowledge transfer and information processing relate to the cognitive processes
that students undergo while taking assessments, other psychological constructs also influence
students’ learning processes and achievement. Personal interest in a certain subject is one such
factor in predicting achievement. Dewey (1913) defined interest as an object, subject, or idea that
becomes an accompanying part of one’s identity. Mitchell (1993) specifies educational interest
as interest directly tied to the content of instruction. The present study will use the same
definition as that of Mitchell (1993).
Students’ interest may also differ based on the different types of problems. Renninger,
Ewen, and Lasher (2002) studied three cases of students with varying levels of math interest and
analyzed how that interest related to math word problems. The researchers found that even when
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word problems are personalized to students’ math abilities, high interest resulted in higher
achievement. Furthermore, higher interest can also result in students rereading problems to
understand their contexts or checking their answers to make sure that they makes sense in the
contexts of the problems (Renninger, Ewen, & Lasher, 2002).
Trautwein, Ludtke, Marsh, Koller, and Baumert (2006) measured interest in ninth grade
students on different mathematics tracks. The researchers found that interest in math is
significantly higher in students in the upper track, but there was little difference in interest for
students in the middle or lower tracks. Interest was also found to be significantly correlated to
both individual achievement on standardized mathematics tests and the overall school
achievement on the same standardized tests (Trautwein, Ludtke, Marsh, Koller, & Baumert,
2006). The same researchers also found marginally significant results stating that there were
reciprocal effects on math interest and self-concept; in other words, rather than one construct
impacting the other, both interest and self-concept impact each other. These results were also
found to be generalizable across gender (Marsh, Trautwein, Ludtke, Koller, and Baumert, 2005).
Other studies have also found personal interest to be related to other psychological constructs.
Ozyurek (2005) found statistically significant correlations between interest in a math class and
self-efficacy, subject preference, previous math performance, and class expectations. These
results were also consistent for undergraduate students who were mathematics majors and not
mathematics majors (Ozyurek, 2005). Therefore, may factors contribute to one’s personal
interest, including math achievement, which is a variable to be used in the current study.
Gender
Gender differences are also important to analyze when it comes to math assessments and
motivation. Much research has been done on how males and females perform on math tests,
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 11
particularly standardized assessments. Liu and Wilson (2009) state that on many large-scale
math assessments, such as the SAT exam, males typically outperform females. However, their
research of the Programme of International Student Assessment (PISA) data in 2000 and 2003
showed that there was no strong correlation between gender and achievement on certain question
types (Liu & Wilson, 2009). Hoffman and Sparatiu (2008) also studied math problem solving
using undergraduate students. They found no significant differences between males and females
in terms of problem solving accuracy and efficiency (Hoffman & Sparatiu, 2008).
Other studies claim that there are significant differences in math achievement between
males and females. A study by Keller (2012) looks at the relationship between gender and
achievement on the SAT and ACT exams, which are nationally used college entrance exams.
Between 1997 and 2010, although the test scores of both males and females slightly increased
over time, males consistently scored higher than females (Keller, 2012). Kaufman, Kaufman,
Liu, and Johnson (2009) found that males scored significantly higher than females in math
achievement on a standardized test, despite the fact that there was no significant difference
between genders regarding fluid or crystallized intelligence. Technology can play a role in both
achievement and motivation between genders. Barkatsas, Kasimatis, and Gialamas (2009) used
the Mathematics and Technology Attitudes Scale (MTAS) to track motivation and attitudes
toward performing math using technology. The study found that students who are comfortable
with technology perform better on classwork and other math assignments. Furthermore, the
results also distinguished between genders. Males were more likely to be on either end of the
motivation and achievement spectrum than females. All students, however, were open to using
technology, and the discussion concluded that technology is a positive mathematical tool
(Barkatsas, Kasimatis, & Gialamas, 2009). Research has even shown that there are differences
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 12
in the strategies that males and females use in different item formats. Katz et al. (2000) examined
the different strategies used for multiple choice and constructed response math items. In this
study, constructed response questions were identical to the multiple choice questions with the
only difference being that there were no choices. Results showed that while item format changed
the difficulty of some questions, the strategies used for different questions did not affect the
achievement. In terms of gender, females tended to use more traditional strategies and scored
better on multiple choice items than constructed response items. Males, on the other hand, used
more nontraditional strategies, but scored about the same on both item formats (Katz et al.,
2000).
Gaps in the Literature
Despite previous research on these topics, there are several gaps in the literature. First,
although Renninger, Ewen, and Lasher (2002) claim that math interest can impact achievement
on different question types, their study was done using a paper-and-pencil assessment. For that
reason, it is worthwhile to study whether similar results will occur with a computer-based
assessment. Second, while there is much research on item format and question type individually
as they relate to achievement on a computer-based test, there is little research comparing the
effects of the interactions between item format and question type on achievement on a computer-
based test. Finally, there is minimal research comparing the relationships among item format,
question type, personal mathematics interest, and gender. As a result, this study will address
several unanswered questions in the literature and attempt to fill the aforementioned gaps.
Research Questions
The following research questions hope to address the gaps in the literature regarding
computer-based math assessments, math interest, and gender:
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1. To what extent do item format (multiple choice or technology-enhanced), question type
(straightforward or word problem), and personal math interest impact algebra
achievement on a computer-based assessment?
2. Are there differences in these effects based on gender?
These research questions will address issues regarding information processing and
knowledge transfer because as students complete a math assessment with different item formats
and question types, they may be required to apply different cognitive processes for each type of
question. If significant differences are found, it may suggest that some question types require
students to use working memory in different ways.
The author hypothesizes that both item format and question type will have an effect on
math achievement, but item format will have a more significant impact. Furthermore, because
the test is computer-based, the researcher hypothesizes that males will have a higher
achievement. This is based on the previous research by Barkatsas, Kasimatis, and Gialamas
(2009). Finally, it is hypothesized that there will be no interaction effect between item format
and gender or between question type and gender.
Method
Participants
The purpose of this study is to examine the relationships between item format, question
type, interest, and gender on math achievement on a computer-based assessment. The
participants of this study will be 225 high school students (n = 225) currently enrolled in an on-
level Algebra II course in a large suburban school district during the 2014-2015 school year.
These students will be selected based on availability and convenience of the researcher. Students
who participate in the study will be enrolled in the high school at which the researcher currently
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 14
teaches. The 225 students will come from a total of nine on-level Algebra II classes, four of
which are taught by the researcher and five of which are taught by other teachers in the same
school. As an incentive, the researcher will randomly select four students who participated in the
study to receive a $25 Target gift card.
The sample will consist of 113 females (n = 113) and 112 males (n = 112). The ages of
the students at the time of the study will range from fifteen years to nineteen years. The mean
age of the students will be 16.34, and the standard deviation will be 0.76. The ethnicity of the
students is expected to be similar to that of the school demographic: About 33% of students will
be Hispanic, 25% of students will be Asian, 24% of students will be white (not of Hispanic
origin), 18% of students will be African-American (not of Hispanic origin), and 2% of students
will be listed as “other.” Approximately twenty students will be categorized as limited English
proficiency. About 55% of students will receive free or reduced lunch on a daily basis.
Approximately 5% of students will be categorized as special education.
Procedures
All students participating in the study will first take a survey in which they self-report
their levels of personal interest in mathematics. The survey consists of four Likert-scale items
and will take approximately one minute to complete. In addition to the Likert-scale items,
students will also record their gender, grade level, and age on the survey. The items of this
survey can be found at the end of this document in Appendix B.
One class period after taking the self-report survey, students will take a measure of math
achievement. Students will have one full class period (90 minutes) to complete this test. Classes
will be randomly assigned to take one of four versions of the math measure: Test A, Test B, Test
C, or Test D. Test A and Test B have the same prompts in the same order. On Test A, the odd
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 15
numbered questions will be multiple choice questions, and the even numbered questions will be
technology-enhanced questions (either fill-in-the-blank or selecting multiple correct answers).
On Test B, the even numbered questions will be multiple choice questions, and the odd
numbered questions will be technology-enhanced questions. Test questions are staggered this
way so that data will be collected for every prompt as both a multiple choice and technology-
enhanced format. On both Test A and Test B, there will be twenty straightforward math
problems (ten multiple choice and ten technology-enhanced) and ten word problem or real-world
application problems (five multiple choice and five technology-enhanced). Creating the
assessments in this way yields four categories: straightforward multiple choice, straightforward
technology-enhanced, word problem multiple choice, and word problem technology-enhanced.
Both tests have the same amount of questions in each category. These tests were created in this
way because the data collected will provide a comparison of the same prompts with different
item formats, as well as a comparison between straightforward questions and word problems.
The items in Test A and Test B can be found at the end of this document in Appendix C.
Test C will have the same exact questions and formats as Test A, but the questions will
be in reverse order. Test D will have the same exact questions and formats as Test B, but the
questions will be in reverse order. Test C and Test D were created to account for test fatigue.
Tyrrell and colleagues (1995) support that visual and mental fatigue can occur as students take
assessments, especially when the assessments are taken on the computer. As students work
through throughout the thirty-question math measure, some may become less motivated or
energized toward the end of the test. Thus, if this occurs, it is important to test whether or not this
occurs in the measure before drawing conclusions about specific test questions or overall
achievement.
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The results of the tests will be available online to the teacher once the test is complete.
Results will include the number of correct answers and the responses to each multiple choice and
technology-enhanced question.
Measures
Math achievement (Mazzarella, 2015). Students will be given a set of thirty math items
on the computer through the program Horizon, which is a commonly used computer-based
assessment program in the county in which the present study will be being conducted. This
program allows teachers to create items of different formats, and teachers will receive the
students’ results for each question when the assessment is complete.
The curriculum used to create these math items aligns with the standards set forth by the
county in which this study will be conducted. These standards also match the standards used to
create the end-of-year state assessment that students in Algebra II are required to take. Some
examples of standards included in the math measure are solving radical, absolute value, and
rational equations, finding the domain and range of various functions, simplifying rational and
radical expressions, identifying properties of a normal distribution, and recognizing and solving
permutations and combinations. According to the state in which the study is taking place, there
are four strands (i.e. standards) that categorize Algebra II test questions: Expressions and
Operations, Equations and Inequalities, Functions, and Statistics (Virginia Department of
Education, 2012). It is also worthwhile to note that students in the county are required to
complete and pass Algebra II in order to graduate. This is important to keep in mind because it
implies that students with a variety of skill levels will be measured in the present study, rather
than only students who choose to enroll in an Algebra II course without it being required.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 17
Personal Math Interest (Mitchell, 1993). The self-report measure will consist of items
measuring personal interest. On this survey, there will be four items measuring personal math
interest. The internal consistency coefficient of this measure was found to be .92, which suggests
that the measure is very reliable (Mitchell, 1993). The survey consists of four Likert-scale items
measuring students’ personal interest in mathematics. One example of an item asks students to
evaluate this statement: “Compared to other subjects, mathematics is exciting to me.” Students
will respond to each item by circling “strongly agree,” “agree,” “slightly agree,” “slightly
disagree,” “disagree,” or “strongly disagree.”
Data Analysis
All data analysis will be conducted using the computer programs SPSS, jMetrik, or
Mplus. The first research question asks: To what extent do item format (multiple choice or
technology-enhanced) and question type (straightforward or word problem) impact students’
math interest and algebra achievement on a computer-based assessment? Several types of
analysis will be used to address this question. First, descriptive statistics will be calculated for
each of the four versions of the mathematics measure. These descriptive statistics, which include
mean and standard deviation, will compare students’ overall achievement on each test. Second,
two separate 2x2 ANOVA tests will be run. The first test will use item format (multiple choice
and technology-enhanced) and question type (straightforward and real-world application) as
independent variables, and math achievement as the dependent variable. The second test will use
the specific technology-enhanced format (fill-in-the-blank and multiple-select) and question type
(straightforward and real-world application) as independent variables, and math achievement as
the dependent variable. For both tests, it is important to consider both the main effects and the
interaction effects.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 18
The second research question asks: Are there differences in these effects based on
gender? To address this question, descriptive statistics, such as mean and standard deviation, will
be calculated. These statistics will compare overall math achievement, achievement on each
question type, achievement on each item format, and interest based on gender. Additionally,
differential item functioning (DIF) will be run on the math items with respect to gender. DIF
analysis is used to determine whether a question is more likely to be answered correctly by one
of two groups (Dimitrov, 2014). In this case, the groups will be classified as male or female. This
analysis will primarily be used to determine whether a question is strongly biased toward one
group. If one group answers a question significantly more accurately than the other, the item may
be recommended to be altered or removed from the test.
In addition to the data analysis that will be run to answer the research questions, there
will also be item analysis run for each question on the math measure to test the psychometric
features of the tests. First, a confirmatory factor analysis will be run to determine if the items in
the math measure align with the four strands that the state outlined. If these strands are distinct,
then further analysis may be done to determine if significant differences exist among the
different strands. Next, classical test theory (CTT) analysis will be run for the multiple choice
questions. Such descriptors in CTT analysis include reliability, which is defined as the ratio of
true score variance to observed score variance; item difficulty, which is defined as the proportion
of examinees who answered the item correctly; and item discrimination, which is defined as the
difference between the proportion of examinees in the upper group who answered the item
correctly and the proportion of examinees in the lower group who answered the item correctly
(Osterlind, 2010). These CTT analyses will be applied to both individual items and the overall
assessment.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 19
Another method of analysis that will be used is Rasch model analysis. Rasch model
analysis can be used to indicate several characteristics about the assessment scores. One such
characteristic is the fit of individual items. Rasch analysis using the jMetrik program will
identify misfit items, which could be an indication of a poor item in general (item fit) or an
anomaly of examinees’ abilities and the items on which they scored correctly or incorrectly
(person fit). Rasch model analysis using jMetrik will also identify the estimated ability level (θ)
of a student given the number of items that they answered correctly (Dimitrov, 2014).
Limitations and Future Research
There are several limitations in regard to this study. One such limitation is the validity of
a self-report survey. It is important to take into account the validity of each measure when
collecting data. In particular, the measure of interest should be carefully examined. Tracey
(2012) claims that single interest scales contain two types of error: systematic error and general
factor variance. Furthermore, these scales do not take into account bias that can influence
students’ interest. For example, interest is often correlated with students’ math scores, which can
cause a problem with validity (Tracey, 2012). For that reason, it is important to analyze the
results of the interest survey to determine whether any of these problems can compromise the
research questions.
Another limitation of this study is the lack of a variable of prior achievement or grade in
high school mathematics. Literature shows that prior achievement does, in fact, impact
motivation and present achievement (Midgley et al., 1989; Trautwein et al., 2006). However, this
variable will not be included in the present study because Algebra I and Geometry are much less
complex courses than Algebra II, and student achievement may vary drastically across these
courses. Furthermore, classroom test grades were not included because some material that is
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 20
taught throughout the year-long Algebra II course is not included in the state assessment, and the
aim of this study is to focus on the implications for the end-of-year exam. Nevertheless, this
study could be improved if the researchers determine a practical and relevant measure of
previous grade to include.
Third, the present study is specific in its content and demographics and, therefore, cannot
be generalized across all subjects or grade levels. In addition to a lack of generalizability to other
content areas, this study may be applicable to only on-level Algebra II. As mentioned, Algebra II
is more complex than Algebra I, and thus requires more skills to achieve mastery. Similarly,
achievement in Geometry requires a different mathematical skill set. Furthermore, this study was
conducted with participants in high school only, and the results cannot be generalized to
elementary or middle school students. Further research is needed to determine whether the
results of this study will be similar across these different demographics and subjects.
Next, the participant selection of this study is that of a convenient sample. The students
of this study attend the school at which the researcher teaches. Some of the participants are even
the researcher’s students. This can create bias and a potential conflict of interest in the research
because of a personal connection to the researcher. To prevent this, the study should be
replicated using students outside of the researcher’s school.
A final limitation of this study may come from the math measure. Although the measure
was created to replicate the state end-of-year exam, the items in this measure will have yet to be
fully analyzed before their initial use in this study. In order to create the most valid results, it is
recommended that the math tests be taken by a different group of students prior to this study, so
that adjustments can be made to invalid or unfair items. If this is not possible, then this study can
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 21
act as a pilot study, and an identical study can be run at a later time with the adjusted math
measures.
Further research could add to the findings of this study in several ways. First, including
additional motivational constructs would give an added element to the study. For example,
measuring students’ task value, self-efficacy, and outcome expectations could provide a wealth
of additional information about students’ motivation as it relates to achievement on computer-
based assessments. Second, although the researcher could not determine an appropriate form of
prior achievement or previous grade to include in this study, further research could use several
different types of prior achievement or previous grade (e.g. previous state test scores) to
determine which are more related to achievement and motivation. Third, future research can also
replicate this study in other subjects or demographics. For instance, it would be worthwhile to
determine if the same results would occur with geometry students or in a different school setting
(e.g. rural or urban).
Educational Implications
The present study contains many practical implications for educational psychology and
secondary mathematics education. Overall, this study may bring to light certain aspects of
computer-based testing that have not previously been analyzed. With more counties and states
changing their standardized testing systems from paper-and-pencil to technology-enhanced, it is
important to investigate whether or not these assessments are testing students accurately and
fairly. The present study will not only determine the differences between item format and
question type, but it will also determine how personal interest will impact achievement.
Additionally, the present study will also determine whether gender is a factor on achievement on
different item formats or question types. It is the hope of the researcher that testing companies
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 22
will take into consideration the findings of this study and make the necessary adjustments to the
tests so that all students have a fair opportunity to succeed. Furthermore, the data of this study
can also provide valuable information to teaching on how to best prepare their students for these
tests.
Whether it is tailoring instruction to certain types of questions or incorporating
motivational strategies to increase interest, the information gathered from this study can help
teachers of math provide the best personalized education for all students. Overall, this study
hopes to sheds light on the relationships among interest, item format, question type, gender, and
math achievement. Through quantitative analysis, significant results may indicate strong
correlations and possible directions for future research. Despite some limitations in the study, the
results will provide insight into some of the complex relationships that exist within educational
psychology.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 23
References
Barkatsas, A., Kasimatis, K., & Gialamas, V. (2009). Learning secondary mathematics with
technology: Exploring the complex interrelationship between students’ attitudes,
engagement, gender and achievement. Computers & Education, 52, 562-570. doi:
10.1016/j.compedu.2008.11.001
Belenky, D. M., & Nokes-Malach, T. J. (2013). Mastery-approach goals and knowledge transfer:
An investigation into the effects of task structure and framing instruction. Learning and
Individual Difference, 25, 21-34. doi: 10.1016/j.lindif.2013.02.004
Burns, M. K., Klingbeil, D. A., & Ysseldyke, J. (2010). The effects of technology-enhanced
formative evaluation on student performance on state accountability math tests.
Psychology in the Schools, 47(6), 582–591. Retrieved from http://mutex.gmu.edu/login?
url=http://search.ebscohost.com/login.aspx?direct=true&db=
ehh&AN=51419288&site=ehost-live
Day, S. B. & Goldstone, R. L. (2012). The import of knowledge export: Connecting findings and
theories of transfer of learning. Educational Psychologist, 47(3), 153-176. doi:
10.1080/00461520.2012.696438
Dewey, J. (1913). Interest and effort in education. Cambridge, MA: Riverside Press.
Dimitrov, D. (2014). Topic 8 – Item Response Theory Part 4: Rasch model- Data fit, item/test
information, and jMetrik Rasch Analysis [PowerPoint Slides]. Retrieved from
https://mymasonportal.gmu.edu.
Fuchs, L. S., Fuchs, D., Stuebing, K., Fletcher, J. M., Hamlett, C. L., & Lambert, W. (2008).
Problem solving and computational skill: Are they shared or distinct aspects of
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 24
mathematical cognition? Journal of Educational Psychology, 100(1), 30-47. doi:
10.1037/0022-0663.100.1.30
Goldhammer, F., Naumann, J., Stelter, A., Toth, K., & Rolke, H. (2014). The time on talk effect
in reading and problem solving is moderated by task difficulty and skill: Insights from a
computer-based large-scale assessment. Journal of Educational Psychology, 106(3), 608-
626. doi: 10.1037/a0034716
Hoffman, B. & Spatariu, A. (2007). The influence of self-efficacy and metacognitive prompting
on math problem-solving efficiency. Contemporary Educational Psychology, 33, 875-
893. doi: 10.1016/j.cedpsych.2007.07.002
Jodoin, M. G. (2003). Measurement efficiency of innovative item formats in computer-based
testing. Journal of Educational Measurement, 40(1), 1-15. doi: 10.1111/j.1745-
3984.2003.tb01093.x
Katz, I. R., Bennett, R. E., & Berger, A. E. (2000). Effects of response format on difficulty of
SAT-mathematics items: It’s not the strategy. Journal of Educational Measurement,
37(1), 39-57. doi: 10.1111/j.1745-3984.2000.tb01075.x
Kaufman, A. S., Kaufman, J. C., Liu, X., & Johnson, C. K. (2009). How do educational
attainment and gender relate to fluid intelligence, crystallized intelligence, and academic
skills at ages 22–90 years? Archives of Clinical Neuropsychology, 24(2), 153–163.
doi: 10.1093/arclin/acp015
Keller, J. (2012). Differential gender and ethnic differences in math performance. Zeitschrift fur
Psychologie, 220(3), 164-171. doi: 10.1027/2151-2604/a000109
Kramarski, B., Weiss, I., & Sharon, S. (2013). Generic versus context-specific prompts for
supporting self-regulation in mathematical problem solving among students with low or
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 25
high prior knowledge. Journal of Cognitive Education and Psychology, 12(2), 197-214.
doi: 10.1891/1945-8959.12.2.197
Liu, O. L., & Wilson, M. (2009). Gender differences in large-scale math assessments: PISA
trend 2000 and 2003. Applied Measurement in Education, 22(2), 164–184. doi:
10.1080/08957340902754635
Marsh, H. W., Trautwein, U., Ludtke, O., Koller, O., & Baumert, J. (2005). Academic self-
concept, interest, grades, and standardized test scores: Reciprocal effects models causal
ordering. Child Development, 76(2), 397-416. doi: 10.1111/j.1467-
8624.2005.00853.x
Martin, J. (2004). Self-regulated learning, social cognitive theory, and agency. Educational
Psychologist, 39(2), 135-145. doi: 10.1207/s15326985ep3902_4
Martinez, M. E. & Bennett, R. E. (1992). A review of automatically scorable constructed-
response item types for large-scale assessment. Applied Measure in Education, 5(2), 151-
169. doi: 10.1207/s15324818ame0502_5
Mayer, R. E. (1995). Learners as information processors: Legacies and limitations of educational
psychology’s second metaphor. Educational Psychologist, 31(3-4), 151-161. doi:
10.1207/s15326985ep3103&4_1
Mayer, R. E. (2011). Applying the science of learning. Boston, MA: Allyn & Bacon.
Mayer, R. E. (2012). Information processing. In Harris, K. R., Graham, S., Urdan, T.,
McCormick, C. B., Sinatra, G. M., & Sweller, J. (Eds.), APA educational psychology
handbook, Vol 1: Theories, constructs, and critical issues. Washington, DC: American
Psychological Association.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 26
Mazzarella, M. (2015). Algebra II Review. Unpublished mathematics assessment, George Mason
University, Fairfax, VA.
Mitchell, M. (1993). Situational interest: Its multifaceted structure in the secondary school
mathematics classroom. Journal of Educational Psychology, 85(3), 424-436. doi:
10.1037/0022-0663.85.3.424
Osterlind, S. J. (2010). Modern measurement: Theory, principles, and applications of mental
appraisal. Boston, MA: Allyn & Bacon.
Ozuru, Y., Best, R., Bell, C., Witherspoon, A. & McNamara, D. S. (2010). Influence of question
format and text availability on the assessment of expository text comprehension.
Cognition and Instruction, 25(4), 399-438. doi: 10.1080/07370000701632371
Ozyurek, R. (2005). Informative sources of math-related self-efficacy expectations and their
relationship with math-related self-efficacy, interest, and preference. International
Journal of Psychology, 40(3), 145-156. doi: 10.1080/00207590444000249
Pomplun, M. & Omar, M. H. (1997). Multiple-mark items: An alternative objective item format?
Educational and Psychological Measurement, 57(6), 949-962. doi:
10.1177/0013164497057006005
Raghubar, K. P., Barnes, M. A., & Hecht, S. A. (2010). Working memory and mathematics: A
review of developmental, individual difference, and cognitive approaches. Learning and
Individual Differences, 20(2), 110-122. doi: 10.1016/j.lindif.2009.10.005
Renninger, K. A., Ewen, L., & Lasher, A. K. (2002). Individual interest as context in expository
text and mathematical word problems. Learning and Instruction, 12, 467-491. doi:
10.1016/S0959-4752(01)00012-3
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 27
Scherer, R & Siddiq, F. (2015). The Big-Fish-Little-Pond-Effect revisited: Do different types of
assessments matter? Computers & Education, 80, 198-210. doi:
10.1016/j.compedu.2014.09.003
Shapiro, E. S. & Gebhardt, S. N. (2012). Comparing computer-adaptive and curriculum-based
measurement models of assessment. School Psychology Review, 41(3), 295-305.
Retrieved from
http://eds.a.ebscohost.com.mutex.gmu.edu/ehost/pdfviewer/pdfviewer?sid=f556a2ef-
3c7d-412d-8d46-bcfc125b71f7%40sessionmgr4001&vid=4&hid=4110.
Stickney, E. M., Sharp, L. B., & Kenyon, A. S. (2012). Technology-enhanced assessment of
math fact automaticity: Patterns of performance for low- and typically achieving
students. Assessment for Effective Intervention, 37(2), 84–94. doi:
10.1177/1534508411430321
Threlfall, J., Pool, P., Homer, M., & Swinnerton, B. (2007). Implicit aspects of paper and pencil
mathematics assessment that come to light through the use of the computer. Educational
Studies in Mathematics, 66(3), 335–348. doi:10.1007/s10649-006-9078-5
Tracey, T. J. G. (2012). Problems with single interest scales: Implications of the general factor.
Journal of Vocational Behavior, 81, 378-384. doi: 10.1016/j.jvb.2012.10.001
Trautwein, U., Ludtke, O., Marsh, H. W., Koller, O., & Baumert, J. (2006). Tracking, grading,
and student motivation: Using group composition and status to predict self-
concept and interest in ninth-grade mathematics. Journal of Educational Psychology, 98(4),
788-806. doi: 10.1037/0022-0663.98.4.788
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 28
Tyrrell, R., Holland, K., Dennis, D., & Wilkins, A. J. (1995). Coloured overlays, visual
discomfort, visual search and classroom reading. Journal of Research in Reading, 181,
10-23.
Virginia Department of Education (2012). Standard of learning (SOL) and testing. Retrieved
from Virginia Department of Education website: http://www.doe.virginia.gov.
Appendix A: Information Processing Theory Model
Figure 1. Basic model of information processing. Retrieved from Mayer (2012).
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 29
Appendix B: Demographics and Interest Measure
Student ID ____________________ DO NOT PUT YOUR NAME ON THIS SURVEY
1. Circle your gender: Male Female
2. Circle your grade: 9 10 11 12
3. Write your age: ___________________
For items 4 – 7, circle the appropriate response based on the statement.
4. Math is enjoyable to me.
Strongly Disagree Disagree Slightly Disagree Slightly Agree Agree Strongly Agree
5. I have always enjoyed studying math in school.
Strongly Disagree Disagree Slightly Disagree Slightly Agree Agree Strongly Agree
6. Compared to other subjects, I feel relaxed studying math.
Strongly Disagree Disagree Slightly Disagree Slightly Agree Agree Strongly Agree
7. Compared to other subjects, math is exciting to me.
Strongly Disagree Disagree Slightly Disagree Slightly Agree Agree Strongly Agree
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 30
Appendix C: Math Measure
Test A
Question 1 :What is the solution set for this equation?
A:
B:
C:
D:
Question 2 :Type your answer into the box. You must give your answer in integer form.
The following sequence is given in recursive form.
What is the value of the fourth term of this sequence?
Question 3 :Which of the following situations involves a permutation?
A: Determining how many different groups of 3 employees can be chosen from 9 employees.
B: Determining how many different ways 7 runners can be assigned lanes on a track for a race.
C: Determining how many different ways to choose 10 students to attend a field trip from a group of 25 students.
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 31
D: Determining how many different ways 4 cashiers can be chosen to work from a group of 7 cashiers.
Question 4 :Click on a box to choose each y-coordinate you want to select. You must select all correct answers.
What are the y-coordinates for the solution to this system of equations?
A: y = -9
B: y = -3
C: y = -2
D: y = 1
E: y = 2
F: y = 6
G: y = 8
H: y = 9
Question 5 :The number of combinations of 7 objects taken 2 at a time is
A: 3B: 7
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 32
C: 21D: 42
Question 6 :Type your answer into the box. You must enter your answer in integer form.
The shoe sizes of a large population are normally distributed with a mean of 8.9 inches and a standard deviation of 0.705 inches. What percentage of the population has a shoe size greater than 9.8 inches? ROUND TO THE NEAREST INTEGER.
Question 7 :
Factor:
A: (2x+3) (3x-7)
B: (2x-3) (3x+7)
C: (3x+2) (2x-7)
D: (3x-2) (2x+7)
Question 8 :Select a box for each correct part of the expression. You must select each correct expression.
Select each part of the simplified expression .
A:
B:
C:
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 33
D:
E:
F:
Question 9 :The area of a triangle varies jointly with the product of the base and the height. A triangle has a base of 12 feet, a height of 3 feet, and an area of 18 square feet. What is the base of a triangle with a height of 4 feet and an area of 36 square feet?
A: 0.5 feet
B: 9 feet
C: 12 feet
D: 18 feet
Question 10 :Type the answer into the box.The number of permutations of 9 objects taken 3 times is
Question 11 :
Which is a solution of ?
A: x = -5
B:x
= C: x = -
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 34
1
D: x = 1
Question 12 :Click on the box to select the value. You must select each correct value.
Two baseballs were thrown on a field as the same time. One ball follows the
path of the function , and the other ball follows the path of
the function , where x is the time in seconds, and f(x) and g(x) are the heights in feet. At what two times, in seconds, are the two balls the same height?
A: 0.7 seconds
B: 1.0 seconds
C: 4.0 seconds
D: 5.14 seconds
E: 5.45 seconds
F: 7.13 seconds
Question 13 :What is the sum of this infinite series?
72 - 36 + 18 - 9 + ...
A: -144
B: -48C: 48
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 35
D: 144
Question 14 :Type your answer into the box. You must enter your answer in integer form.
Let and , what is ?
Question 15 :A new rollercoaster at an amusement park follows the path of the
function , where x is the time, in seconds, after the rollercoaster begins, and f(x) is the height of the rollercoaster, in yards. Between which two times, in seconds, is the rollercoaster increasing in height?
A:
B:
C:
D:
Question 16 :Select each expression that is equivalent. You must select all correct expressions.
Identify each expression that is equivalent to 1.
A:
B:
C:
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 36
D:
E:
F:
G:
H:
Question 17 :
Which of the following describes the end behavior of as x approaches infinity?
A: y approaches negative infinity
B: y approaches -2
C: y approaches 3
D: y approaches infinity
Question 18 :Type your answer into the box. You must enter your answer in integer form.
The heights of Galapagos penguins are normally distributed with a mean of 49 cm and a standard deviation of 1.82 cm. If a scientist measures the heights of 300 penguins, how many penguins are expected to be between 48.4 cm and 50.1 cm tall? ROUND YOUR ANSWER TO THE NEAREST INTEGER.
Question 19 :Which is the equation of an asymptote of the graph of the following equation?
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 37
A: x = -3
B: y = 3
C: x = 6
D: y = 6
Question 20 :Click on the box to select an interval. You must select each correct interval.
Indicate each intervals where the graph is only increasing.
A:
B:
C:
D:
E:
F:
G:
H:
Question 21 :A math class consists of 10 girls and 8 boys. The teacher wants to
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 38
choose 2 girls and 2 boys to go on a trip. How many different groups could the teacher choose?
A: 73
B: 146
C: 1260
D: 5040
Question 22 :Type your answer into the box. You must enter your answer in integer form.
If y varies directly with the square root of x, what is the constant of variation if y = 36 when x = 9?
Question 23 :
What are the zeros of the function ?
A: x = -16 and x = 0
B: x = -16, x = -8, and x = 2
C: x = -8 and x = 2
D: x = -2 and x = 8
Question 24 :Click on the box to select the correct equation. You must select each correct equation.
A baseball was thrown by a player, and hit the ground after exactly 5 seconds. If x represents the time in seconds and y represents the height of the ball, which of the following functions could represent the path of the
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 39
ball?
A:
B:
C:
D:
E:
F:
Question 25 :A normally distributed set of numbers has a mean of 75 and a standard deviation of 7.97. What percentage of values lies between 70 and 85?
A: 11%B: 37%C: 63%D: 89%
Question 26 :Click on the correct box to select each value. You must select each correct value.
The domain of the function is all real numbers except -
A: -7B: -4C: -3D: 0E: 3
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 40
F: 4G: 7
Question 27 :In a school, 12 students are running for 4 class officers- a president, a vice president, a secretary, and a treasurer. If each position is to be held by one person and no person can hold more than one position, in how many ways can those positions be filled?
A: 48
B: 495
C: 11880
D: 20736
Question 28 :Click on each box to select each function. You must select each correct function.
Which of the following functions are in the same family as the
function ?
A:
B:
C:
D:
E:
F:
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 41
Question 29 :The steps to simplify an expression are shown below:
Step 1: 4(x+3) - 3x + 1Step 2: 4x + 12 - 3x + 1Step 3: 4x - 3x + 12 + 1Step 4: x + 13
Which of the following properties justifies getting from Step 2 to Step 3?
A: Associative Property
B: Commutative Property
C: Distributive Property
D: Transitive Property
Question 30 :Type your answer into the box. ROUND YOUR ANSWER TO THE NEAREST TENTH.
A baseball player throws a ball from one end of the field to the other. A fan measures the path of the ball and determines that it follows the
function , where x is the time in seconds and f(x) is the height in feet. What is the maximum height of the ball, in feet? ROUND YOUR ANSWER TO THE NEAREST TENTH.
Test B
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 42
Question 1 :Type your answer into the box. Enter your answer as a whole number.
What is the solution set for this equation?
Question 2 :The following sequence is given in recursive form.
What is the value of the fourth term of this sequence?
A: 29B: 33C: 61
D: 125
Question 3 :Click on the box to choose the situation. You must select all correct situations.
Which of the following situations involve a permutation?
A: Determining how many different ways to choose 3 employees from a group of 9 employees.
B: Determining how many different ways 7 runners can be assigned lanes on a track for a race.
C: Determining how many different seating charts can be made placing 6 people around a table.
D: Determining how many 5-letter passwords can be made using the
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 43
word "graph."
E: Determining how many different groups of 10 students can be chosen to go on a field trip from a group of 25 students.
F: Determining how many different ways 4 cashiers can be chosen to work from a group of 6 cashiers.
Question 4 :What are the y-coordinates for the solutions to this system of equations?
A: y = 1 and y = 9
B: y = -3 and y = -9
C: y = -2 and y = 6
D: y = 2 and y = 8
Question 5 :Type the answer into the box.
The number of combinations of 7 objects taken 2 at a time is
Question 6 :The shoe sizes of a large population are normally distributed with a mean of 8.9 inches and a standard deviation of 0.705 inches. What percentage of the population has a shoe size greater than 9.8 inches? Round to the nearest integer.
A: 5%B: 10%C: 20%
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 44
D: 34%
Question 7 :Click on a box for each factor you want to select. You must select all correct factors.
Select all of the factors of:
A: (2x+3)
B: (2x-3)
C: (3x+2)
D: (3x-2)
E: (2x+7)
F: (2x-7)
G: (3x+7)
H: (3x-7)
Question 8 :
Which of the following expressions is equivalent to ?
A:
B:
C:
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 45
D:
Question 9 :Type your answer into the box. You must enter your answer in integer form.
The area of a triangle varies jointly with the product of the base and the height. A triangle has a base of 12 feet, a height of 3 feet, and an area of 18 square feet. What is the base of a triangle with a height, in feet, of 4 feet and an area of 36 square feet?
Question 10 :The number of permutations of 9 objects taken 3 times is
A: 27B: 84
C: 504
D: 729
Question 11 :Click on each solution to the equation. You must select each correct solution.
Select all the solutions of .
A: x = -5
B:x
=
C: x = -1
D: x = 1
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 46
E:x
=
F: x = 5
Question 12 :Two baseballs were thrown on a field as the same time. One ball
follows the path of the function , and the other ball
follows the path of the function , where x is the time in seconds, and f(x) and g(x) are the heights in feet. At what two times, in seconds, are the two balls the same height?
A: 0.70 seconds and 5.45 seconds
B: 7.13 seconds and 5.14 seconds
C: 0.70 seconds and 7.13 seconds
D: 5.14 seconds and 5.45 seconds
Question 13 :Type your answer into the box.
What is the sum of this infinite series?
72 - 36 + 18 - 9 + ...
Question 14 :
Let and . What is ?
A: -29B: 35C: 75
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 47
D: 152
Question 15 :Click on the box to select the interval. You must select each correct interval.
A new rollercoaster at an amusement park follows the path of the
function , where x is the time, in seconds, after the rollercoaster begins, and f(x) is the height of the rollercoaster, in yards. Between which two times, in seconds, is the rollercoaster increasing in height?
A:
B:
C:
D:
E:
F:
Question 16 :Which expression is equivalent to 1?
A:
B:
C:
D:
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 48
Question 17 :Type your answer into the box. You must give your answer in integer form.
What value does y approach in the function as x approaches infinity?
Question 18 :The heights of Galapagos penguins are normally distributed with a mean of 49 cm and a standard deviation of 1.82 cm. If a scientist measures the heights of 300 penguins, how many penguins are expected to be between 48.4 cm and 50.1 cm tall? Round your answer to the nearest integer.
A: 84
B: 107
C: 168
D: 204
Question 19 :Click on each box to choose each asymptote. You must select all correct asymptotes.
Which are the equations of the asymptotes of the graph of the following function?
A: x = -3
B: y = -3
C: x = 3
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 49
D: y = 3
E: x = 6
F: y = 6
Question 20 :
Throughout which of the following intervals is only increasing?
A:
B:
C:
D:
Question 21 :Type your answer into the box. You must enter your answer in integer form.
A math class consists of 10 girls and 8 boys. The teacher wants to choose 2 girls and 2 boys to go on a trip. How many different groups could the teacher choose?
Question 22 :If y varies directly as the square root of x, what is the constant of variation if y = 36 and x = 9?
A: 1.5
B: 2C: 4D: 12
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 50
Question 23 :Click on the box to select the zeros. You must select each correct zero.
Which of the following are zeros of the function ?
A: x = -16
B: x = -8
C: x = -2
D: x = 0
E: x = 2
F: x = 8
Question 24 :A baseball was thrown by a player, and hit the ground after exactly 5 seconds. If x represents the time in seconds and y represents the height of the ball, which of the following functions could represent the path of the ball?
A:
B:
C:
D:
Question 25 :Type your answer into the box. You must enter your answer in
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 51
integer form.
A normally distributed set of numbers has a mean of 75 and a standard deviation of 7.97. What percentage of values lies between 70 and 85? ROUND TO THE NEAREST INTEGER.
Question 26 :
The domain of the function is all real numbers except -
A: -7, -4, 4
B: -7, 4
C: -4, 7
D: 4
Question 27 :Type your answer into the box. You must enter your answer in integer form.
In a school, 12 students are running for 4 class officers- a president, a vice president, a secretary, and a treasurer. If each position is to be held by one person and no person can hold more than one position, in how many ways can those positions be filled?
Question 28 :Which of the following is in the same family as the
function ?
A:
B:
C:
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 52
D:
Question 29 :Click on the box to select a property. You must select each correct property.
The steps to simplify an expression are shown below:
Step 1: 4(x+3) - 3x + 1Step 2: 4x + 12 - 3x + 1Step 3: 4x - 3x + 12 + 1Step 4: x + 13
Which of the following properties justify Step 2, Step 3, and Step 4?
A: Associative Property
B: Commutative Property
C: Distributive Property
D: Inverse Property
E: Substitution Property
F: Transitive Property
Question 30 :A baseball player throws a ball from one end of the field to the other. A fan measures the path of the ball and determines that it
follows the function , where x is the time in seconds and f(x) is the height in feet. What is the maximum height of the ball, in feet?
A: 2 feet
ITEM FORMAT AND QUESTION TYPE ON MATH ACHIEVEMENT 53
B: 5.4 feet
C: 6 feet
D: 9.2 feet